Clearly, in order to be certified as undead, you need a death certificate. Which means the Witch King doesn't qualify, since, like an old soldier, he didn't die, he just faded away.

Clearly this is open to interpretation, I can say that in no way is a wraith "alive" and thus they've crossed over into a form of "un-death" - beings who can corporally effect the living long after they've stopped taking breath. The undead are not merely zombies or vampires, folks. The Witch-King most certainly qualifies as "undead" in my mind and so do the barrow-wights.

A strong argument could be made that the Nazgul are animated bodies, feeding off of and controlled by a powerful supernatural spirit. They no longer feel emotions like a human, don't eat like a human, don't age as a human. Their existence depends on the strength and presence of a separate supernatural entity. They are merely extensions of this supernatural force and the will and personalities of who they once were are gone. Who they once were is dead. King Arthur: Who are you who can summon fire without flint or tinder?